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METHODS TO CALCULATE ROOTS OF EQUATIONS

NUMERIC METHODS IN ENGINEERING

DUBAN CASTRO FLOREZCOD:2073091

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TAYLOR´S SERIES

• It is written the it formulates. • Those are made derived until the n order. • It is replace those derived in the function (being replaced

the xi value). • It is written f(xi+1)= f(x).

MACLAURIN: Pnx• Same Taylor procedure, but xi=0 • The answer is given as Pn(xi+1)=Pn(x). to Remember:

n=#derivatives. • To calculate the true error

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METHODS FOR THE CALCULATE OF ROOTS

1.MÉTHOD OF BISECTIÓN

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This in the first interval

0)(.)( ri xfxfIf This in the second interval

• It is continuous until the xr value is approximate to the real value of the function, determined by the tolerance given for:

•It is graph the function, the cut is looked for with axis x, xi and xs are determined.

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2.FALSE POSITION

The same procedure is continued that the bisection method, to exception of the change in the form of finding xr.

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3.FIXED POINT

• To the function f(x) you him supreme x to find g(x) and with the error you can determine if the function converges.

• The xi value is determined knowing the graph of the function (Mathematics of

Microsoft or Excel).

• When it is xi with g(xi), g(xi)=xi+1, that is to say it will be the value to evaluate.

• If the function doesn't converge the method of clearances of x it is used, the times

that it is necessary until the approximate value of g(x) it is similar to that of the f(x root).

• The first one is evaluated derived in the point to evaluate, if it is gives smaller than

1, it converges, if it is gives bigger it diverges.

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4.NEWTON - RAPHSON

Having f(x)=0 to calculate the one derived of the function symbolically. To choose an initial value. To find by means of it formulates it of Newton-Raphson. To calculate% Ea. If% Ea =% is then the result of the root it is reported, otherwise to make that the one

calculated is the new one and to return to the step 3.

The interval is determined to study for half graphic, approaching a value xi to the root, and finding the tangent in that point.

This value xi is replaced in f(xi) and in f'(x), it stops then to use the it formulates of up.

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5.SECANT

• They should know each other two positions xi-1 and xi for then to replace in it formulates it:

• To remember that xi>xi-1 • When he/she is xi+1, now that it will be my xi and xi-1 it will be

the xi that had previously.• The necessary xi is calculated until the value is approximate to

the real value of the root of the function given.

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BIBLIOGRAPHY

•CHAPRA, Steven C. y CANALE, Raymond P.: Numérics Méthods for Engineers. McGraw Hill 2002.

•Presentations of teacher’s class(Eduardo Carrillo).

•SANTAFE, Elkin R. “Introducción a la Simulación Numérica de Yacimientos”. Clases-Universidad Industrial de Santander Año-2008.